⛛f(r) = ?
f’ r/|r|
⛛ ᐧ (a × b) or ⛛ × (a ᐧ b)
use vector expansion
⛛𝜙 = ?
ex d𝜙/dx + ey d𝜙/dy + ez d𝜙/dz
d 𝜙 = ?
(⛛𝜙)dr = (t ⛛𝜙) ds
what is rate of change of 𝜙 along t
t ⛛𝜙
surface normal from ⛛𝜙
⛛𝜙 / |⛛𝜙|
when is parameterisation needed
make x = x(t)
F(r) = Fx(r) ex + Fy(r) ey + Fz(r) ez
div formula
⛛ ᐧ F = dFx/dx + dFy/dy + dFz/dz
curl formula
⛛xF = (dFz/dy - dFy/dz)ex +
(dFx/dz - dFz/dx)ey +
(dFy/dx - dFx/dy)ez
laplacian formula
⛛²𝜙 = d²𝜙/dx² + d²𝜙/dy² + d²𝜙/dz²
line integral formula
when is line integral conservative and what is the formula
integral from t1 to t2 F(r(t)) dr/dt dt = ϕ(B)−ϕ(A)ifconservative
dr = (dx/d0, dy/d0, dz/d0)d0
Fx = d𝜙/dx
Fy = d𝜙/dy
Fz = d𝜙/dz
green’s theorem
integralc Pdx + Q dy = integralD (dQ/dx - dP/dy)dA
conservative vector field definition
integralr F dr = integralr ⛛𝜙 dr = 0
Fdr is exact differential
integral F dr is path independent
surface / flux integral formula
∫∫s FdS = ∫∫s F n dS = ∫y∫x Fn dxdy = ∫∫∫v ⛛ ᐧ F dV if conservative
= F n Area if uniform F
surface / flux integral formula if conservative
∫∫∫v ⛛ ᐧ F dV
surface / flux integral formula if uniform F
F n Area
divergence theorem
∫∫∫v ⛛ ᐧ F dV = ints F dS
C is the boundary of S which is a closed curve
stokes’ theorem
integrals (⛛ x F)dS = integralc F dr
⛛ x F = 0 <=> F is conservative field
state orientation by RHR (anticlockwise wrt normal).
flux integral for sphere
∫∫s F r^ r² sin 0 d0 d𝜙 = ∫∫s F x/r r^2 sin 0 d0 d𝜙
flux integral for cylinder
sum 3 surfaces separately
volume integral formula
∫∫∫f(rcos, rsin, z) r dr d dz
∫v (⛛ᐧF) dV
∫dV F ᐧ dS
